Question

Let $X$ be a set with exactly $5$ elements and $Y$ be a set with exactly $7$ elements. If $\alpha$ is the number of one-one functions from $X$ to $Y$ and $\beta$ is the number of onto functions from $Y$ to $X$, then the value of $\frac{1}{5!}(\beta -\alpha )$ is _______.

Answer 1

$n\left(X\right)=5$
$n\left(Y\right)=7$
$\Rightarrow \alpha \to$ Number of one-one function ${=}^7{C}_5\times 5!$
$\Rightarrow \beta \to$ Number of onto function $Y$ to $X$
$1,1,1,1,31,1,1,2,2$
$\Rightarrow \frac{7!}{3!4!}\times 5!+\frac{7!}{(2!{)}^{3}3!}\times 5!={(}^7{C}_3+{3.}^7{C}_3)5!=4{\times }^7{C}_3\times 5!$
$\Rightarrow \frac{\beta -\alpha }{5!}=4{\times }^7{C}_3{-}^7{C}_5=4\times 35-21=119$.